Properties

Label 3234.2.a.bd
Level $3234$
Weight $2$
Character orbit 3234.a
Self dual yes
Analytic conductor $25.824$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.8236200137\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} - 2 q^{5} + q^{6} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{3} + q^{4} - 2 q^{5} + q^{6} + q^{8} + q^{9} - 2 q^{10} - q^{11} + q^{12} + (\beta - 4) q^{13} - 2 q^{15} + q^{16} - 2 q^{17} + q^{18} + ( - 3 \beta - 2) q^{19} - 2 q^{20} - q^{22} + (2 \beta - 2) q^{23} + q^{24} - q^{25} + (\beta - 4) q^{26} + q^{27} + ( - 4 \beta + 4) q^{29} - 2 q^{30} + (3 \beta - 2) q^{31} + q^{32} - q^{33} - 2 q^{34} + q^{36} + ( - 2 \beta - 2) q^{37} + ( - 3 \beta - 2) q^{38} + (\beta - 4) q^{39} - 2 q^{40} + (4 \beta - 6) q^{41} + (2 \beta - 2) q^{43} - q^{44} - 2 q^{45} + (2 \beta - 2) q^{46} + ( - 7 \beta - 2) q^{47} + q^{48} - q^{50} - 2 q^{51} + (\beta - 4) q^{52} + (6 \beta - 2) q^{53} + q^{54} + 2 q^{55} + ( - 3 \beta - 2) q^{57} + ( - 4 \beta + 4) q^{58} + (2 \beta - 4) q^{59} - 2 q^{60} + ( - \beta - 4) q^{61} + (3 \beta - 2) q^{62} + q^{64} + ( - 2 \beta + 8) q^{65} - q^{66} + ( - 2 \beta - 4) q^{67} - 2 q^{68} + (2 \beta - 2) q^{69} - 4 \beta q^{71} + q^{72} + (4 \beta - 6) q^{73} + ( - 2 \beta - 2) q^{74} - q^{75} + ( - 3 \beta - 2) q^{76} + (\beta - 4) q^{78} + (6 \beta - 8) q^{79} - 2 q^{80} + q^{81} + (4 \beta - 6) q^{82} + (5 \beta + 2) q^{83} + 4 q^{85} + (2 \beta - 2) q^{86} + ( - 4 \beta + 4) q^{87} - q^{88} + ( - 3 \beta - 8) q^{89} - 2 q^{90} + (2 \beta - 2) q^{92} + (3 \beta - 2) q^{93} + ( - 7 \beta - 2) q^{94} + (6 \beta + 4) q^{95} + q^{96} - 7 \beta q^{97} - q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 4 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 4 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} - 4 q^{10} - 2 q^{11} + 2 q^{12} - 8 q^{13} - 4 q^{15} + 2 q^{16} - 4 q^{17} + 2 q^{18} - 4 q^{19} - 4 q^{20} - 2 q^{22} - 4 q^{23} + 2 q^{24} - 2 q^{25} - 8 q^{26} + 2 q^{27} + 8 q^{29} - 4 q^{30} - 4 q^{31} + 2 q^{32} - 2 q^{33} - 4 q^{34} + 2 q^{36} - 4 q^{37} - 4 q^{38} - 8 q^{39} - 4 q^{40} - 12 q^{41} - 4 q^{43} - 2 q^{44} - 4 q^{45} - 4 q^{46} - 4 q^{47} + 2 q^{48} - 2 q^{50} - 4 q^{51} - 8 q^{52} - 4 q^{53} + 2 q^{54} + 4 q^{55} - 4 q^{57} + 8 q^{58} - 8 q^{59} - 4 q^{60} - 8 q^{61} - 4 q^{62} + 2 q^{64} + 16 q^{65} - 2 q^{66} - 8 q^{67} - 4 q^{68} - 4 q^{69} + 2 q^{72} - 12 q^{73} - 4 q^{74} - 2 q^{75} - 4 q^{76} - 8 q^{78} - 16 q^{79} - 4 q^{80} + 2 q^{81} - 12 q^{82} + 4 q^{83} + 8 q^{85} - 4 q^{86} + 8 q^{87} - 2 q^{88} - 16 q^{89} - 4 q^{90} - 4 q^{92} - 4 q^{93} - 4 q^{94} + 8 q^{95} + 2 q^{96} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
1.00000 1.00000 1.00000 −2.00000 1.00000 0 1.00000 1.00000 −2.00000
1.2 1.00000 1.00000 1.00000 −2.00000 1.00000 0 1.00000 1.00000 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(1\)
\(11\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3234.2.a.bd yes 2
3.b odd 2 1 9702.2.a.cy 2
7.b odd 2 1 3234.2.a.bc 2
21.c even 2 1 9702.2.a.ci 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3234.2.a.bc 2 7.b odd 2 1
3234.2.a.bd yes 2 1.a even 1 1 trivial
9702.2.a.ci 2 21.c even 2 1
9702.2.a.cy 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3234))\):

\( T_{5} + 2 \) Copy content Toggle raw display
\( T_{13}^{2} + 8T_{13} + 14 \) Copy content Toggle raw display
\( T_{17} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 8T + 14 \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 4T - 14 \) Copy content Toggle raw display
$23$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$29$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$31$ \( T^{2} + 4T - 14 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$41$ \( T^{2} + 12T + 4 \) Copy content Toggle raw display
$43$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$47$ \( T^{2} + 4T - 94 \) Copy content Toggle raw display
$53$ \( T^{2} + 4T - 68 \) Copy content Toggle raw display
$59$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T + 14 \) Copy content Toggle raw display
$67$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$71$ \( T^{2} - 32 \) Copy content Toggle raw display
$73$ \( T^{2} + 12T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} + 16T - 8 \) Copy content Toggle raw display
$83$ \( T^{2} - 4T - 46 \) Copy content Toggle raw display
$89$ \( T^{2} + 16T + 46 \) Copy content Toggle raw display
$97$ \( T^{2} - 98 \) Copy content Toggle raw display
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